\chapter{Diffusion flames}

\section{Description of nonpremixed flames}

\begin{figure}%
\centering
\includegraphics[width=\textwidth]{turbflamelengthhottel.jpg}%
\caption{Development of diffusion flame structures with increase in nozzle velocity \cite{Hottel1949}}%
\label{fig:turbflamelength}%
\end{figure}

A nonpremixed flame or a diffusion flame has, in contrast to a premixed flame, a separate fuel and oxidizer stream. It is called a diffusion flame since the reaction and flame speed is characterized by the rate of diffusion between fuel and oxidizer. It is the most common type of combustion in industrial applications. A typical diffusion flame is encountered in gas turbines, but it is also used for many other industrial applications. Fig. \ref{fig:turbflamelength} shows the development of flame length with increase in nozzle velocity. Turbulent flames are observed to have the same flame length when the jet velocity is increased. This indicates the rate of turbulent mixing is proportional with jet velocity.

Some important dimensionless constants for nonpremixed flames are given below

The Schmidt number ($Sc$) is the relation between the viscosity of the flow and the mass diffusivity

\begin{equation}
Sc=\frac{\nu}{\mathcal{D}}
\label{eq:Schmidt number}
\end{equation}

The Prandtl number ($Pr$) provides a measure of the relative contribution of momentum and energy transport by diffusion.
\begin{equation}
Pr=\frac{\nu}{\alpha}
\label{eq:Prandtl number}
\end{equation}

The Lewis number ($Le$) describes the ratio of thermal diffusivity over mass diffusivity. It couples the energy equation with the mass transport equation

\begin{equation}
Le=\frac{Sc}{Pr}=\frac{\lambda}{\rho c_p \mathcal{D}}=\frac{\alpha}{\mathcal{D}}.
\label{eq:Lewis number}
\end{equation}


\subsection{Mixture fractions and the conserved scalar}

The mixture fraction $\xi$ is a property of the mixture can be defined based on the element mass fractions. The element mass fractions can not be changed by reaction, they are only changed by mixing, therefore the mixture fraction is a conserved scalar.

\begin{equation}
\xi=\frac{Z_i-Z_{i,2}}{Z_{i,1}-Z_{i,2}}
\label{eq:Mixfrac}
\end{equation}

Here $1$ and $2$ denote the fuel and oxidizer stream respectively. For $\xi=1$ the mixture consists of fuel only, and for $\xi=0$ the mixture consists of oxidizer only. $\xi$ is a conserved during chemical reaction and therefore it is often called a \emph{conserved scalar}. $\xi$ does not have a chemical source or sink term. 

The transport equation for the element mass fraction is

%Transport equation for element mass fraction Z_i
\begin{equation}
	\frac{\partial (\rho Z_i )}{\partial t}+\frac{\partial }{\partial {{x}_{j}}}(\rho {{u}_{j}})=\frac{\partial }{\partial {{x}_{j}}}(\rho \mathcal{D} \frac{\partial Z_i }{\partial {{x}_{j}}})
\label{eq:Ztransport}
\end{equation}

The similar governing equation for the mixture fraction is then
%The governing equation for the mixture fraction or conserved scalar

\begin{equation}
	\frac{\partial (\rho \xi )}{\partial t}+\frac{\partial }{\partial {{x}_{j}}}(\rho \xi {{u}_{j}})=\frac{\partial }{\partial {{x}_{j}}}(\rho \mathcal{D} \frac{\partial \xi }{\partial {{x}_{j}}})
\end{equation}

With some simplifications we can ignore the source term in the energy equation. With the assumption that the Lewis number equals 1 ($\alpha=\mathcal{D}$), and by ignoring radiation heat transfer and heat dissipation, and by ignoring pressure gradients we can write

\begin{equation}
\xi=\frac{h-h_2}{h_1-h_2}.
\label{eq:mixfrac by h}
\end{equation} 

Since the mixture fraction is a conserved scalar we can now write

\begin{equation}
	\tilde{\phi }=\int_{0}^{1}{\phi (\xi )f(\xi )d\xi }
\label{eq:pdfaverage}
\end{equation}

where $f(\xi)$ is a probability density distribution based on $\xi$.

We can get the average properties in the turbulent flame by using this PDF (Probability Density Function). For example

\begin{equation}
\overline{T}=\int_0^1{T(\xi) f(xi) d \xi}
\label{eq:Taveragepdf}
\end{equation}

\begin{equation}
\overline{\rho Y_k}=\int_0^1{\rho(\xi) Y_k(\xi) f(\xi) d \xi}
\label{eq:rhoY pdf}
\end{equation}

The Reynolds equation in mixture fraction space reads

%Reynolds equation in mixture fraction space.
\begin{equation}
\frac{\partial}{\partial t}(\rho \overline{\xi})+\frac{\partial}{\partial x_j}(\rho \overline{\xi} \overline{u}_j)=\frac{\partial}{\partial x_j}=(\rho \mathcal{D} \frac{\partial \overline{\xi}}{\partial x_j}-\rho \overline{\xi' u'_j})
\label{eq:mixfrac Reynolds}
\end{equation}

And the scalar variance is given as

%Scalar variance
\begin{equation}
\widetilde{{\xi''}^2}=\frac{\overline{\rho {\xi''}^2}}{\overline{\rho}}
\label{eq:xivar}
\end{equation}

Which gives eq \ref{eq:Scalar dissipation}.
%Scalar dissipation rate
\begin{equation}
\overline{\rho \chi}=2 \overline{\rho \mathcal{D} \frac{\partial \xi}{\partial x_j}\frac{\partial \xi''}{\partial x_j} } 
\approx 2 \overline{\rho \mathcal{D}  \left( \frac{\partial \xi''}{\partial x_j} \right)^2 } 
\approx 2 \overline{\rho \mathcal{D}  \left( \frac{\partial \xi}{\partial x_j} \right)^2 }
\approx 2 \overline{\rho \mathcal{D}} \left(\frac{\partial \tilde{\xi}}{\partial x_j} \right)^2
\label{eq:Scalar dissipation}
\end{equation}

Here $\mathcal{D}$ is the mixture diffusivity. When modelling turbulent combustion a common assumption is equal diffusivity of species. The last part of \ref{eq:Scalar dissipation} has been used to model the scalar dissipation in \cite[]{warnatz2006combustion} and \cite{Stanciu2001}, but there is no mathematical proof for the last transition. The assumption that this transition is valid is made in order to approximate the scalar dissipation from available data. The scalar dissipation rate, $\chi$ has the unit $s^{-1}$ and quantifies the rate of molecular mixing, and is a key property of the diffusion flame, since the diffusion flame is controlled by the rate of mixing. $\chi$ is very difficult to measure, therefore it is usally modeled. $\mathcal{D}$ in \ref{eq:Scalar dissipation} measures the rate of diffusion for the mixture fraction. In turbulent flows the rate of turbulent mixing is much greater than the rate of molecular mixing. The rate of mass diffusion and the rate of thermal diffusion are related to the same turbulent eddies. For turbulent diffusion flames a common assumption is $\alpha \approx \mathcal{D}$ and $Le=\alpha / \mathcal{D} \approx 1$. 


\section{The prescribed PDF method}
\label{sec:prescribed-PDF}

A commonly used method of modeling turbulent combustion is the statistical approach using probability density functions. A \emph{probability density function} (PDF) gives the probability distribution of a certain property of the reacting flow at a given location in the flame. The probability distribution is given by the mean and variance of the mixture fraction at each point of the flame. By introducing probability density functions we decouple the properties from the spatial dimensions of the flame and thus there is no need to solve the a complicated turbulent flow field. This greatly simplifies the calculations. The spatial dimensions are preserved through Eq. \ref{eq:mixfrac Reynolds}.

%Mixture fraction approach is only valid for fast chemistry
\begin{figure}%
\includegraphics[width=\textwidth]{pdfdistribution.jpg}%
\caption{Schematic shapes of PDFs in a turbulent jet flame \cite{versteeg2007introduction}}%
\label{fig:pdfdistr}%
\end{figure}

The shape of the pdf has to be prescribed. A frequently used formulation is the $\beta$-distribution Eqs. \ref{eq:beta-func pdf}-\ref{eq:beta parameters}. According to Ertesvåg \cite[p.162]{Ertesvåg2000} and Warnatz, Maas and Dibble \cite[p.214]{warnatz2006combustion} a clipped Gaussian function, as used by \cite{Stanciu2001}, probably fits better, but the need to calculate a tedious exponential term and a non-continuous probability density function makes this method more difficult to use. The shape of the $\beta$-PDF depends on the values of the parameters $a$ and $b$ which must both be positive ($a>0$ and $b>0$). Swaminathan and Bilger (1999) \cite{Swaminathan1999} did DNS calculations of PDFs in turbulent combusting flows. Their study supports the use of a $\beta$-PDF to model turbulent flames.

The $\beta$-distribution is given by the following equations
%Beta distribution
\begin{equation}
f(\xi)=\frac{\xi^{a-1}(1-\xi)^{b-1}}{B(a,b)}
\label{eq:beta-func pdf}
\end{equation}

where $0 \le \xi \le 1$. Here B is the $\beta$-function

%Beta function

\begin{equation}
B(a,b)=\int_0^1{x^{a-1}(1-x)^{b-1}dx}=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
\label{eq:Beta function}
\end{equation}

\begin{equation}
\overline{\xi}=\frac{a}{a+b} \text{ and }  \overline{\xi'^2}=\frac{\overline{\xi}(1-\overline{\xi})}{1+a+b}
\label{eq:beta parameters}
\end{equation}

Solving for $a$ and $b$ gives

\begin{equation}
a=\overline{\xi} \left(\frac{\overline{\xi}(1-\xi)}{\overline{\xi''^2}} -1 \right) \text{ and } b=(1-\overline{\xi}) \frac{a}{\overline{\xi}}
\label{eq:beta exponents}
\end{equation}


\subsection{Relations for fast chemistry}

\begin{figure}%
\centering
\includegraphics[width=0.5\textwidth]{Y(xi).jpg}%
\caption{Distribution of mass fractions in a diffusion flame with the fast chemistry assumption}%
\label{eq:Y(xi)}%
\end{figure}

The relations for fast chemistry are given as follows

\begin{equation}
Y_{fu}(\xi)=\begin{cases} 
0 &\text{ for } 0\le \xi \le \xi_{st} \\
(Y_{fu})_1\left(\frac{\xi-\xi_{st}}{1-\xi_{st}} \right)	 & \text{ for } \xi_{st} <\xi<1 \\
\end{cases}									  
\label{eq:Fast fuel mass fraction}
\end{equation}

\begin{equation}
Y_{ox}(\xi)=\begin{cases} 
0 &\text{ for } 0\le \xi \le \xi_{st} \\
(Y_{ox})_2\left(\frac{\xi-\xi_{st}}{1-\xi_{st}} \right)	 & \text{ for } \xi_{st} <\xi<1 \\
\end{cases}									  
\label{eq:Fast oxidizer mass fraction}
\end{equation}

\begin{equation}
Y_{in}(\xi)=(Y_{in})_1 (1-\xi)
\label{eq:Fast inert mass fraction}
\end{equation}

\begin{equation}
Y_{prod}(\xi)=1-Y_{fu}(\xi)-Y_{ox}(\xi)-Y_{in}(\xi)
\label{eq:Fast product mass fraction}
\end{equation}

\begin{equation}
T(\xi)=\begin{cases}
			\xi T_1+(1-\xi ) T_2 + \frac{Q}{c_p} (Y_{fu})_1 \xi  & \text{ for } 0 \le \xi \le \xi_{st} \\ 
 			\xi T_1+(1-\xi ) T_2 + \frac{Q}{c_p} (Y_{fu})_1 \xi_{st} \frac{1-\xi }{1-\xi_{st}} & \text{ for } \xi_{st} \le \xi \le 1
\end{cases}
\label{eq:FastTmixfrac}
\end{equation}

From Eq. \ref{eq:FastTmixfrac} it is seen that the derivative of the temperature is a constant in each of the two intervals.

\begin{equation}
\frac{dT}{d\xi}=\begin{cases}
			\xi T_1-T_2 + \frac{Q}{c_p} (Y_{fu})_1  & \text{ for } 0 \le \xi \le \xi_{st} \\ 
 			\xi T_1-T_2 - \frac{Q}{c_p} (Y_{fu})_1 \frac{\xi_{st}}{1-\xi_{st}} & \text{ for } \xi_{st} \le \xi \le 1
\end{cases}
\label{eq:FastderivT}
\end{equation}


\subsection{Modeling of entropy production}
These equations are listed from \cite{Ertesvåg2004}, but a variation of these relations can also be found in \cite{Stanciu2001}.

Sum of local entropy production terms in a flame
\begin{equation}
(\dot{\overline{S}}'''_{gen})_{tot}=(\dot{\overline{S}}_{gen})'''_{chem}+(\dot{\overline{S}}_{gen})'''_{Q}+(\dot{\overline{S}}_{gen})'''_{mass}+(\dot{\overline{S}}_{gen})'''_{visc}
\label{eq:total entropy prod}
\end{equation}

The reaction rate as a function of mixture fraction and scalar dissipation rate is given as
\begin{equation}
R_k= -\frac{1}{2} \rho \chi \frac{\partial^2 Y_k}{\partial \xi^2}
\label{eq:reaction rate}
\end{equation}

The chemical reaction contribution to entropy production is given as

\begin{multline}
(\dot{\overline{S}}'''_{gen})_{chem}= - \overline{\frac{1}{T} \sum_{k}{\mu_k R_k}}
\\ 
= \overline{\rho} \tilde{\chi} \frac{(Y_{fu})_1}{2(1-\xi_{st})} \frac{(\mu_{fu}(\xi_{st})+ r \mu_{ox}(\xi_{st}) - (1+r)\mu_{pr}(\xi_{st}))}{T(\xi_{st})} f(\xi_{st}).
\label{eq:chementropy}
\end{multline} 

The heat flux contribution is

\begin{multline}
(\dot{\overline{S}}'''_{gen})_{Q}=\overline{-\frac{q_j}{T^2}\frac{\partial T}{\partial x_j}}=\overline{\frac{\lambda}{T^2} \left(\frac{\partial T}{\partial x_j} \right)^2}
= \overline{\frac{\lambda}{T^2} \left(\frac{\partial T}{\partial \xi} \right)^2 \left(\frac{\partial \xi}{\partial x_j} \right)^2} \\
\approx \overline{\frac{\lambda}{2 \mathcal{D}} \frac{\rho}{(\rho T)^2} \left(\frac{dT}{d \xi} \right)^2 \rho \chi}.
\label{eq:Qentropy}
\end{multline}


The contribution of mass transfer to Eq. \ref{eq:total entropy prod} is given as

\begin{multline}
(\dot{\overline{S}}'''_{gen})_{mass}= \overline{\sum_k{ \left(- \frac{j_{k,j}}{T} \right) \left(\frac{\partial \mu_k}{\partial x_j} \right)_T}} 
\approx \sum_k{\mathcal{R}_k \overline{\left(- \frac{j_{k,j}}{Y_k} \right) \frac{\partial Y_k}{\partial x_j}}} \\
= \sum_k{\mathcal{R}_k \overline{ \frac{\rho \mathcal{D}}{Y_k} \left(\frac{\partial Y_k}{\partial x_j} \right)^2}}
= \sum_k{\mathcal{R}_k \overline{ \frac{1}{2 Y_k} \left(\frac{d Y_k}{d \xi} \right)^2} \rho \chi} \\
= \frac{1}{2} \overline{\rho} \tilde{\chi} \sum_k{\mathcal{R}_k \int_0^1{\frac{1}{Y_k} \left( \frac{dY_k}{d \xi} \right)^2 f(\xi) d\xi}}.
\label{eq:massentropy}
\end{multline}




The dissipation term of the entropy equation can be split in a mean-flow term and a turbulence term. The entropy production from viscous dissipation in the mean entropy equation is given as
\begin{equation}
(\dot{\overline{S}}'''_{gen})_{visc}=\overline{\left(\frac{\Phi}{T} \right)}=\overline{\frac{\tau_{ij}}{T}} \frac{\partial \tilde{u}_i}{\partial x_j}+\overline{\frac{\tau_{ij}}{T} \frac{\partial u_i ''}{\partial x_j}}=\overline{\frac{\tau_{ij}}{T}} \frac{\partial \tilde{u}_i}{\partial x_j} + \left(\frac{\rho \epsilon}{T} \right).
\label{eq:viscous entropy production}
\end{equation}


\section{The Flamelet Model}
A turbulent flame can be modeled as a series of flat laminar flames or flamelets. A program like FlameMaster \cite{FlameMaster} creates a flamelet library for a given set of parameters.

In \cite{Ertesvåg2000} the flamelet regime is described as the part of a Borghi-diagram where the turbulence intensity is low and the flame front is thinner than the Kolmogorov length scale, which is the smallest turbulent length scale. The flame front is rucked and looks like a series of small laminar flames or flamelets. Each of these flamelets are modeled as one dimensional laminar flames, and is coupled with the chemical kinetics mechanisms which are defined for laminar flames. \cite{PitschFlamelet} has more information about the flamelet method.

FlameMaster \cite{FlameMaster} can create flamelet libraries for an opposing jet flame. The program has built in routines for solving $H_2$ and $CH_4$ as fuel. A good description of how to create flamelet libraries is given in \cite{PitschFlamelet} and \cite[p.388]{versteeg2007introduction}.

%The steady flamelet equations

%Species
%
%\begin{equation}
%\rho \frac{\chi}{2} \frac{\partial^2 Y_k}{\partial \xi^2}+R_k=0
%\label{eq:flamelet species}
%\end{equation}
%
%Temperature
%\begin{equation}
%\rho \frac{\partial T}{\partial t}- \rho \frac{\chi}{2} \frac{\partial^2 T}{\partial \xi^2} -\frac{1}{c_p}\frac{\partial p}{\partial t}+\sum_k{\frac{h_k}{}}
%\label{eq:flamelet temperature}
%\end{equation}
%

